What is the Definition and Mathematical Explanation of Density of States?

[Greg Bernhardt]

Definition/Summary

This term most commonly refers to the number of quantum states having energy within a given small energy interval divided by that interval.

Equations

$$g(E)=\sum_{s}\delta(E-E_s)$$
$$N=\int dE g(E)$$

The "density of states" need not (but it most often does) refer to states per energy interval. For example, for free particles in a box of volume $\mathcal{V}$, the density of states for a given wavevector $\mathbf{k}$ (rather than energy) is a constant:
$$g_{\mathbf{k}}=\frac{\mathcal{V}}{{(2\pi)}^3}.$$
The above equation is the basis for the well-known replacement
$$\sum_{\mathbf{k}}(\ldots)\to\int \mathcal{V}\frac{d^3 k}{{(2\pi)}^3}(\ldots)$$

Extended explanation

The density of states
$$g_{\mathbf{k}}=\frac{\mathcal{V}}{{(2\pi)}^3}\;,$$
results from applying periodic boundary conditions to free waves in a box of volume $\mathcal{V}$ and counting. Thus
$$\delta N = d^3 k g_{\bf k}=d^3 k\frac{\mathcal{V}}{{(2\pi)}^3}\;.$$

If the energy E only depends on the magnitude of $\mathbf{k}$, E=E(k), then we may also write
$$\delta N = d k k^2 \frac{4\pi \mathcal{V}}{{(2\pi)}^3}<br /> =<br /> \frac{4\pi\mathcal{V}}{{(2\pi)}^3}dE \frac{k^2}{v}\equiv dE g(E)\;,$$
where
$$v=\frac{dE}{dk}\;,$$
is the velocity.

For the case where momentum is carried by particles with an effective mass $m^*$ we have
$$k=m^*v\;,$$
and
$$g(E)=\frac{4\pi \mathcal{V}}{{(2\pi)}^3}m^*\sqrt{2 E m^*}\;.$$

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[AaronS]

Thanks for the overview of density states!